Problem 38 - Elementary and Intermediate Algebra

Question

Let \(f(x)=2 x+1\) and \(g(x)=x^{2}-1 .\) Find each of the following. $$ (g \circ f)(2) $$

Step-by-Step Solution

Answer

The value of \((g \circ f)(2)\) is 24.

Step 1: Understand the Notation

The notation \((g \circ f)(2)\) represents the composition of functions \(g\) and \(f\) evaluated at \(x = 2\). This means we first need to find \(f(2)\) and then substitute that result into \(g(x)\).

Step 2: Evaluate f at x=2

To find \(f(2)\), substitute \(x = 2\) into the function \(f(x) = 2x + 1\). This gives \(f(2) = 2(2) + 1 = 4 + 1 = 5\).

Step 3: Substitute into g

Now that we have \(f(2) = 5\), substitute this value into the function \(g(x) = x^2 - 1\). Compute \(g(5) = 5^2 - 1 = 25 - 1 = 24\).

Step 4: Write the Final Answer

After substituting and evaluating, the result of \((g \circ f)(2)\) is 24.

Key Concepts

Evaluating Functions

Evaluating functions involves substituting a specific value for the variable in a function and calculating the result. Think of it as putting an input into a machine and getting an output. For example, consider the function \( f(x) = 2x + 1 \). If we evaluate \( f \) at \( x = 2 \), it means we replace \( x \) in the function with \( 2 \). This gives:

• \( f(2) = 2(2) + 1 \)
• which simplifies to \( 4 + 1 = 5 \)

When you evaluate a function, ensure to follow the order of operations: perform multiplications and divisions before additions and subtractions. Be precise with your calculations, as small errors can lead to incorrect results.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and arithmetic operations. They are like sentences of numbers and symbols combined using mathematical rules. For example, \( 2x + 1 \) is an algebraic expression where \( 2x \) means \( 2 \) multiplied by \( x \), and everything is added by \( 1 \).When dealing with algebraic expressions, know how to:

• Identify terms: Each separate part of an algebraic expression. In \( 2x + 1 \), \( 2x \) and \( 1 \) are terms.
• Understand coefficients: These are numbers multiplying the variable. In \( 2x \), \( 2 \) is the coefficient.
• Apply arithmetic operations: Ensure to use operations correctly to manipulate expressions correctly for evaluation.

Learning these aspects can help you to easily work with and simplify various expressions.

Function Operations

Function operations involve performing mathematical operations such as addition, subtraction, multiplication, division, and composition on functions. In this context, function composition is a key operation.Composition of functions allows you to apply one function to the results of another function. Represented as \((g \circ f)(x)\), it means applying \( f(x) \) first, and then using that result in \( g(x) \). It's like a two-step process:

• Step 1: Find \( f(x) \) for a certain \( x \). Example: \( f(2) = 5 \).
• Step 2: Use the result from Step 1 in \( g(x) \). Example: \( g(5) = 24 \).

Order matters in function composition. Depending on whether \( f \) or \( g \) is applied first, the results can differ. Practicing this operation helps in solving complex expressions by breaking them into simpler parts, one function at a time.